Algebraic Property Testing

Transcription

1 .. Symmetries in Algebraic Property Testing by Elena Grigorescu Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of ARCHNVES Doctor of Philosophy in Computer Science and Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY MASSACHU-sTT SINS TITUTE OF TOV '1LOGy OCT J September 2010 Massachusetts Institute of Technology L All rights reserved. Lj2RARIES Author... Department of Electrical Engineering and Computer Science August 16, 2010 Certified by Madhu Sudan Fujitsu Professor of Electrical Engineering and Computer Science Thesis Supervisor <: -I /) A ccepted by... Professor Terry P. Orlando Chair, Department Committee on Graduate Students

2 Symmetries in Algebraic Property Testing by Elena Grigorescu Submitted to the Department of Electrical Engineering and Computer Science on August 16, 2010, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science and Engineering Abstract Modern computational tasks often involve large amounts of data, and efficiency is a very desirable feature of such algorithms. Local algorithms are especially attractive, since they can imply global properties by only inspecting a small window into the data. In Property Testing, a local algorithm should perform the task of distinguishing objects satisfying a given property from objects that require many modifications in order to satisfy the property. A special place in Property Testing is held by algebraic properties: they are some of the first properties to be tested, and have been heavily used in the PCP and LTC literature. We focus on conditions under which algebraic properties are testable, following the general goal of providing a more unified treatment of these properties. In particular, we explore the notion of symmetry in relation to testing, a direction initiated by Kaufman and Sudan. We investigate the interplay between local testing, symmetry and dual structure in linear codes, by showing both positive and negative results. On the negative side, we exhibit a counterexample to a conjecture proposed by Alon, Kaufman, Krivelevich, Litsyn, and Ron aimed at providing general sufficient conditions for testing. We show that a single codeword of small weight in the dual family together with the property of being invariant under a 2-transitive group of permutations do not necessarily imply testing. On the positive side, we exhibit a large class of codes whose duals possess a strong structural property ('the single orbit property'). Namely, they can be specified by a single codeword of small weight and the group of invariances of the code. Hence we show that sparsity and invariance under the affine group of permutations are sufficient conditions for a notion of very structured testing. These findings also reveal a new characterization of the extensively studied BCH codes. As a by-product, we obtain a more explicit description of structured tests for the special family of BCH codes of design distance 5.

3 Thesis Supervisor: Madhu Sudan Title: Fujitsu Professor of Electrical Engineering and Computer Science

4 Acknowledgments I have been extremely fortunate to be Madhu's advisee - clearly! I am indebted to him for his constant support, encouragements and guidance, for his openness and availability, and for introducing me to so many beautiful research directions. It goes without saying that I have learned a lot about doing research from Madhu, and what I really like the most is the importance of asking daring questions; and being always optimistic about the proximity to a solution; and making collaboration enjoyable, and inclusive; and more. Thanks Madhu for all these, and I wish that we will collaborate again. I loved working on the topics of this thesis with Tali. Her friendliness and kind advice have been really helpful and motivational. I am thankful to my committee members Ronitt Rubinfeld and Piotr Indyk for agreeing to read my thesis and for being so accommodating with setting up my defense date. I have also benefited from Ronitt's support and collaboration starting early in my graduate program. Even though we haven't worked together on research, Piotr's charisma and witty sense of humor have always been refreshing. Yet again, I have been lucky to collaborate with some wonderful colleagues on topics that have not been included in this thesis, and I am grateful to each of them: (in random order) Sofya Raskhodnikova, Madhav Jha, Victor Chen, Kyomin Jung, Arnab Bhattacharyya, Ronald de Wolf, Ronitt Rubinfeld, Irit Dinur, Swastik Kopparty, Ning Xie, David Woodruff, Asaf Shapira, Jakob Nordstr6m. At different stages in my graduate student life it has been great to have around, share thoughts, plans, advice and support with my colleagues and friends from the lab: (in no particular order) Tasos Sidiropoulos, Eddie Nikolova, Victor Chen, Alex Andoni, Brendan Juba, Angelina Lee, Shubhangi Saraf, Swastik Kopparty, Sergey Yekhanin, David Woodruff, Ben Rossman, Prasant Gopal, Jakob Nordstr6m, Arnab Bhattacharyya, Kevin Matulef, Krzysztof Onak, Kyomin Jung, Ning Xie, Mihai Patra cu, Petar Maymounkov.

5 Joanne Hanley's readiness to help with all the administrative issues has made things so much easier for me all these years, and Be Blackburn's sweets corner has always been a nice procrastination destination. I am also grateful to many people who had a positive impact on my academic development before coming to MIT. To mention only a few of them, I thank Joe Gallian for his trust and guidance during his Research Experience for Undergraduates program, where I got a first taste of research. My undergraduate Math and CS professors from Bard College: Lauren Rose, Bob McGrail, Ethan Bloch, Rebecca Thomas and Sven Anderson encouraged me in all my endeavors. I am also extremely indebted to my Math and Physics professors from high-school: Constantin Grigoriu, Dorel Haralamb, and Camelia Neta, whose impeccable teaching styles and interest for scientific rigor were exemplary. I thank all my friends for being such a great and spirited company, for inventing distractions from research and for keeping me in touch with the world outside the lab. Mom, Dad, Miha and Tanveer, your role in my life is beyond words!

6 Contents 1 Introduction 1.1 Property testing and algebraic property testing Structure and symmetry in testing A few motivating questions O ur results A counterexample to the AKKLR conjecture Explicit structured testing in some BCH codes Sufficient conditions for structured testing Organization and credits Bibliographic notes Testing various properties Symmetries and testing Preliminaries Algebraic properties Error-correcting codes Property testing and locally testable codes Invariance

7 3 Description of Cyclic/Affine Invariant Linear Families 3.1 Cyclic-invariant families Affine-invariant families Cyclic/affine codes as polynomial ideals Bibliographic notes Transitivity is Insufficient for Local Testability 4.1 The conjecture The counterexample, basic properties, and proof ideas. 4.3 Proof of main theorem Reed-Muller of Order d families Key lemma Putting it together D iscussion Explicit Structured Testing in Common Algebraic Families 5.1 Definitions and main result Sufficient conditions for single orbit A warm-up: explicit single orbit for the ebch(1, n) Explicit single orbit for the ebch(2, n) Explicit single orbit for RM(d, n) Succinct Representation of Codes with Applications to Testing 6.1 Main results and implications Implications to property testing Implications to BCH codes

8 Overview of techniques and helpful lemmas.. Proofs of the helpful lemmas Proofs of the main theorems Analysis of the cyclic case Analysis of the affine-invariant case.. On using results from additive number theory D iscussion Conclusions and Future Directions 7.1 Towards a full characterization of affine invariant codes Further related work in testing non-linear, linear-invariant properties A Missing details from Chapter 5

9 List of Figures 1-1 Relations among the families considered in this work

10 List of Tables 6.1 Comparison between Weil-Carlitz-Uchiyama and Bourgain bounds.

11 Chapter 1 Introduction Decision problems occur ubiquitously in computation, and while very often they might be hard to solve exactly, many models have been proposed to approach reasonably efficient relaxations. In Property Testing, the model of our focus, the goal is to distinguish between objects that belong to a class, and objects that differ in many locations from each object in the class. A tester algorithm in this model is allowed to accept objects that do not belong to the class, but which are close to some object in the class. The algorithm should use randomness and run in sub-linear time, hence it should make a correct decision after reading only a restricted, but judiciously chosen portion of the input. Such so-called local algorithms are desirable in settings where the decision problem might not be known to be efficiently computable, but also even when it admits polynomial (super-linear) time algorithms. Surprisingly, many natural questions have been shown to admit local algorithms with good error parameters. A domain where local algorithms are both desirable and computationally feasible is in protecting information from errors. In practice for example, data storage devices use redundant encoding in order to facilitate recovery from errors. However, when the amount of error is too large, recovery could become too time consuming, and in fact unaffordable. Local algorithms to test membership in error correcting codes might be employed in these scenarios in order to quickly decide what data is recoverable.

12 The focus of this thesis is on testing membership in error correcting codes. Error correcting codes can be viewed as a special subclass of algebraic families of functions. The ultimate driving goal of this research is to understand features that distinguish between algebraic families for which membership can be tested in a time-efficient manner, and those which provably require non-local algorithms. This direction is strongly motivated by intimate connections with Locally Testable Codes and Probabilistically Checkable Proofs, and has received wide attention in recent years. 1.1 Property testing and algebraic property testing A brief history Blum, Luby and Rubinfeld [33] initiated the field of Property Testing by proposing a tester for the class of linear functions. The immediate followup works of Rubinfeld and Sudan [89], Babai, Fortnow and Lund [15], and Babai, Fortnow, Levin and Szegedy [14], considered testing polynomials of higher degrees in various settings of parameters and domains. Algebraic property testing rapidly took off ever since, and has so far found myriads of applications across theoretical computer science. These results were instrumental in the proofs of MIP=NEXP [15] and in the highly acclaimed PCP Theorem of Arora, Lund, Motwani, Sudan and Szegedy [12]. Testing graph properties has also been well-studied. Graph property testing was introduced by Goldreich, Goldwasser and Ron in [53] who considered properties such as colorability, bipartiteness or connectedness in the dense graph model. The recent results of Alon et al. [5] and Borgs et al. [35] completely characterized testable properties in this model. Property testing Formally, a tester for a property P is a randomized algorithm which can access a given object given as a black-box via queries. Based on the answers to the queries, the algorithm makes an accept/reject decision which satisfies the following conditions: if the object belongs to P it should accept; otherwise, if the

13 object is far from P it should reject with high probability over the randomness of the algorithm. P is thought of as a collection of properties, i.e. P = UnPs for n -* o, where the objects in P, have size n. The tester is local if the number of queries it performs does not depend on the size of the input object, that is, the number of queries is independent of n. The notion of distance to P is captured by metrics dependent on the descriptions of the objects belonging to the property. When the property refers to sets of graphs, distance is measured in the number of edges to be added to or removed from the input graph in order to build a graph in the target collection. In this thesis we concentrate on algebraic properties, namely collections of functions f : D -± R, mapping a vector space D over a field, into a subfield R. In this case the notion of distance is given by the Hamming metric, which is the number of places the function needs to be modified in order to obtain a function from the target family. A tester could be adaptive if the choice of the queries it makes depends on the answers it had received; it is non-adaptive otherwise. The definition above describes a single sided tester; when the test may also err on objects in the family it is called double sided. Locally Testable Codes Locally Testable Codes (LTCs) form a special class of algebraic testable properties. They are error correcting codes for which membership can be tested with a small number of queries. Error correcting codes are usually described by sets of vectors, called codewords, such that the Hamming distance between pairwise vectors is somewhat large. Their design parameters are the block length N and alphabet E. An alternate view of a code C C EN is as collections of functions in {D -± R}, where D = N and R = E (i.e. if c E C the function c(a) = c, Va E D.) The set of linear functions, and the set of low degree polynomials are some basic examples of LTCs. Building LTCs with good parameters has been a direction of active research [57, 22, 24, 44, 84], bearing strong connections with similar trends in the study of PCPs. In this thesis we focus on a second intriguing direction in the area, namely on understanding what features of codes/algebraic properties could reveal global structure from an average local structure. The current trends in the

14 study of LTCs have been recently surveyed in [21]. 1.2 Structure and symmetry in testing A starting example In coding theoretic terms, testing whether a function is linear [33] corresponds to testing membership in the Hadamard code, formally denoted by H = {fa : IF' - IF 2 fa(x) = aix 2, a E IF}. i=o To describe a tester for this family one uses the evident fact that for any a, # E Fn every linear function f E H satisfies the constraint f(a) + f(#) + f(a + /) n- 0. A tester for linearity simply picks uniformly at random a, # E Fn and queries f at a, # and a +. It accepts if and only if f(a) + f() + f(a + ) = 0, and rejects otherwise. An alternate way to represent this test is as a binary vector of weight 3, indexed by elements of Fn and supported on a, #, a + #. This vector has the property that it has inner product 0 with each codewords of H. This view will become useful in analyzing future testers. Consider now the set of all binary vectors whose inner product is 0 with each codeword in H. As we will see later on, each such vector could represent a test for the Hadamard code, and its support size is called the locality of the test. This set of vectors forms a vector space H'. Moreover, this set characterizes the Hadamard code, in the sense that no other function g : F- F 2, has inner product 0 with each function in H'. Formally, H' is the dual of the Hadamard code, i.e. the Hamming code. Linearity and Duality The example above illustrates the concepts we will be working with in this thesis. We focus on linear families of functions 1, namely families for which if f, g E P then f + g C P. Viewed as a collection of vectors, a linear family is just a vector space. Notice that the Hadamard and Hamming codes are 'Not to be confused with families of linear functions.

15 linear codes. Each linear family can be associated with a unique dual family, which is the vector space dual to it. In other words, the dual of P C {F -- F 2 }, denoted P', is {g : F IF 2, EZ f(x)g(x) = 0, Vf e P}. The dual family is of central interest in testing linear properties, since essentially, every test (even in the adaptive or double-sided setting) must belong to it [23]. This point has motivated the need for a better understanding of the structural features of the dual family that are relevant to testing. Our work delves into this connection with the goal of identifying necessary and sufficient conditions for local testability. Symmetry in algebraic properties The structural features of a family can be studied from the perspective of the set of symmetries that the family exhibits. The initial systematic study by Kaufman and Sudan [72] on the role of symmetries in algebraic property testing has sparked a wave of great interest in this connection [60, 61, 74, 25, 54, 31, 30]. A group of symmetries or invariances acting on a family P C { f:d R} is a set of functions i : D -± D such that f E P if and only if f o w E P. When dealing with codes, it is most common to only consider functions w which are permutations, and hence f o -F is a permuted codeword. The largest group of symmetries acting on a family is called the automorphism group. 2 Many common families of algebraic properties exhibit large automorphism groups. Invariance under linear transformations of the domain is a most commonly encountered symmetry. The Hadamard code, and Reed Muller codes are invariant under the linear group {g : F' - F' I g(x) = Ax, A c F"X"} (a.k.a GL(n, 2)). Invariance under affine transformations occurs also commonly. For example, Reed Muller codes are invariant under the affine group {g : F F"xn, b E Fn} (a.k.a AGL(n, 2))3. _ F- I g(x) = Ax + b, A c 2In the coding literature, the set of all permutation that keep the code invariant is called the permutation group. The automorphism group of a code includes, besides permutations, transformations that multiply each element of a function by a non-zero element of the field. In this thesis we only focus on binary functions, and thus the two groups coincide. In a few places we slightly abuse its common usage by calling it the automorphism group when we only mean permutation group. 3 Strictly speaking, AGL(n, 2) and GL(n, 2) only refer to nonsingular transformations A. We are

16 In this thesis we concentrate on linear and affine groups of symmetries and analyze the testability of codes that feature these invariances. 1.3 A few motivating questions To summarize our introductory exposition, in this work we investigate the relations among (1) bases of low weight vectors, (2) affine/linear invariance and (3) testability. In this section we propose a few basic questions tackling the interplay between these notions. We describe our results in more detail in Section 1.4. Codes with/without bases of low weight Ben-Sasson et al. [23] formalized the fact that any tester (even adaptive or double-sided ones) for membership in a linear family F can be reduced to picking g c FL. That is, the tester queries a given function f at locations in the support of g. This result motivated understanding the testability of families whose duals are generated by bases of low weight functions. Bases of low weight functions have been very relevant in testing many common families. Returning to our starting example of the Hadamard code, one can prove that the Hamming code is generated by the weight 3 codewords, i.e by the minimum weight codewords. Similarly, bases of low weight functions were used in testing Reed Muller codes [7, 63] and dual-bch codes [67]. However, the existence of a low weight arbitrary basis is not sufficient for testing [23]. Random Low Density Parity Check (LDPC) codes, although generated by weight 3 codewords, require a large number of queries in order to test for membership. Vaguely speaking, this phenomenon is caused by the fact that large collections of low weight functions cannot combine into (sum up to) another low weight function, which would be a necessary condition. Understanding when a family can be characterized by low weight functions is an important step in understanding its testability. This task could be highly non-trivial: abusing notation slightly.

17 given an arbitrary basis possibly of large weight vectors is there a small weight basis for it? What if the family is represented by collections of functions, or say polynomials? What if the family exhibits some large group of symmetries? We note that techniques to analyze this type of questions are rare in the literature: a successful approach was made in [23], by means of analyzing properties of expander graphs. This leads us to a first question that motivates out work. Question 1 Can one exhibit explicit families that are invariant under a large group of symmetries, contain low weight functions but cannot be characterized by low weight functions? That is, can one exhibit such a family for which any basis must contain a large weight vector? The family we exhibit for an answer to this question will also be relevant to showing our negative testing results. In particular, we prove that the dual of an affine subcode of Reed Muller codes of order 2 cannot have a low-weight basis (See Chapter 4.) This result has applications to a conjecture of Alon et al. [7] (the AKKLR conjecture), which attempted to unify specific testing approaches in the literature. Namely, the conjecture stated that families that are '2-transitive' and admit small constraints should be testable by tests with small locality. The AKKLR [7] conjecture has been so far an important source of sufficient conditions for testability in the literature, motivating the major contributions that the work of Kaufman and Sudan [72] has brought in this direction. The single orbit characterization in common codes Kaufman and Sudan [72] realized that the symmetries of a family can lead to more structured bases and furthermore to structured tests. In particular, they focused on families that can be completely characterized by a single function and its set of transformations under a group of symmetries. This property is denoted by single orbit characterization under a group of symmetries of the family.

18 Again, the Hadamard code provides an example of families that admit single orbit characterization. Indeed, every codeword of the Hamming code can be obtained as a linear combination of the set of permutations AGL(n, 2) of the codeword supported on (ei, e 2, ei + e 2 ) (here ei and e 2 are the standard basis vectors in Fn.) This property similarly holds for Reed Muller codes. For different settings of field size versus degree, a single orbit characterization exists, but its description might differ with the setting. For example, for Reed Muller codes of order d in fields of characteristic 2 (that is, RM(d) = {f : -F F 2 deg(f) < d},) a single orbit characterization of the dual is given by a function supported at {Span(ao, a 2,..., ad) + #}, for some ao,...,c ad,# IF". For degree d < IKI and RM(d) {f : K' -± K, deg(f) < d} a single orbit generator for the dual is supported at {1, w,..., wd+1}, where w is a primitive element of K. The single orbit property is to some extent expected under such large groups of symmetries (i.e. AGL(2, n)): one should able to find a basis of dimension, say O(2n), among 2n2 vectors. A more intriguing question is whether such single orbit bases could be found for smaller groups of invariances, which leads to a 2nd sequence of questions that we propose. Question 2 Do Reed Muller codes have the single orbit property under a smaller group of symmetries? Also do BCH codes (which are also extensively studied codes) have the single orbit characterization property? If so, what is the smallest group under which this property holds for BCH codes? We show that Reed Muller codes (in characteristic 2) have the single orbit property under even smaller groups of invariances, namely under affine groups over a domain 1F 2. (a.k.a. AGL(1, 2n)) (see Chapter 5). Furthermore, we show that this property also holds for BCH codes. In some cases, BCH codes have this property under an even smaller group of invariances, i.e. under linear transformations of F 2 - (a.k.a. GL(1, 2n)) (see Chapter 6).

19 Structured testing in general settings The reason why the single orbit characterization is a relevant feature is due to the fact that it immediately leads to a notion of structured testing. A structured tester simply picks a random permutation from a group of invariances and computes its composition with a low weight generator of the single orbit. Kaufman and Sudan [72] showed that having the single orbit property under affine invariant transformations implies structured testing. Structured tests are nice since as soon as a single low weight generator is known, the rest of the tests are immediately explicitly specified by a group of symmetries. This observation prompts us to another set of questions of broad interest. Question 3 What general families of functions admit structured tests under the affine group? What general families have the single orbit property under other groups? To answer these questions we show some general families that have the single orbit characterization under affine and linear invariances (see Chapter 6.) We proceed with a more detailed account of our results. 1.4 Our results We focus on families F C {IF 2 n -+ F2} that are invariant under affine/cyclic transformations of IF 2. and show positive and negative testing results A counterexample to the AKKLR conjecture As mentioned, our first result provides an answer to Question 1 discussed above and a counterexample to the AKKLR conjecture (in Chapter 4.) We provide an explicit family F C {F2n - F 2 } that is invariant under a group of 2-transitive transformations, but that cannot be tested with a small number of queries. This counterexample is a subcode of Reed Muller codes of order 2, and in addition, it is affine invariant. We argue that any small weight vector in F' must belong to the

20 dual of the Reed Muller code of order 2. Hence any basis for the dual of F must have a large weight codeword. For an illustration of the broader context of this example see Figure transitivity Informally, a family of functions is 2-transitive if any two coordinates look the same as any other two. This notion of symmetry is related to that of pairwise independence, which in turn plays a crucial role in the analysis of selfcorrectors for linear properties. To be more precise, a self-corrector for a function f (which is assumed to be correct on most inputs) computes the value of f at each location x, with high probability, from the value of the function at a few other places. Self-correctors and testers for linear properties have very similar features. While local testers use functions of small weight in the dual to test for membership, self-correctors use the same dual-functions in order to correct corrupted locations in the given function. A common analysis argument on the success probability of self correctors (e.g. [33, 7, 70]) relies on the fact that f(x) can be computed from values f(xi) such that x and x are almost pairwise independent. The fact that 2-transitive codes with small dual distance are correctable was formalized in [73]. These apparent interconnections between 2-transitivity, correctability and testability prompted Alon et al. [7] to propose 2-transitivity as an indicator for local testability. Finally, we note that the most natural 2-transitive groups are the affine groups, and in fact families that are 2-transitive but not affine invariant are non-trivial to construct. Supporting evidence The proposed sufficient conditions were an initial attempt at a more unified theory of the features that enable testability in Hadamard, Reed Muller, and BCH codes. A confirmation of the conjecture in a broader context was exhibited in [72]. Their results show that the existence of a low weight dual function together with invariance under the affine group implies the existence of a special basis (the single orbit characterization), which in turn implies testability.

21 As additional evidence from the negative side, the conjecture does not hold for random LDPC codes. Clearly such codes do not have large groups of symmetries, and in particular they are not invariant under a 2-transitive group. As shown by Ben-Sasson et al. [23] random LDPC codes require a large number of queries in order to test for membership, even though their dual may contain small weight functions (in fact they may contain a basis of small weight functions.) Implications Disproving the conjecture is a step toward a better understanding of the structural properties that enable testing in algebraic settings. Our results here can be stated as saying that for families F C { K " --+ F} where KI - oc (in our case m = 1,) affine invariance and the existence of a small weight dual function does not imply local testing. This contrasts with the case when K is of fixed size [72]. Even though the conjecture is false in general, it has the merit of identifying symmetry (2-transitivity,) as a possible indicator of testability. This view has inspired positive results in the same vein and has been expanded in subsequent works, including this thesis Explicit structured testing in some BCH codes Chapters 5 and Chapter 6 give partial answers to Question 2 described above. In Chapter 5 we start our study of the single orbit characterization in BCH codes, by presenting an explicit structured basis for the restricted family of BCH codes of design distance 5. We also show that Reed Muller codes can be generated by an explicit single orbit under the group AGL(1, 2"). Furthermore, the results of Chapter 6 imply that general BCH codes can be generated by a single low weight codeword under the affine group AGL(1, 2 n), for some settings of the field size. Moreover, BCH codes admit low weight single orbit generators even for slightly smaller groups, namely under the linear (cyclic) group GL(1, 2').

22 Explicit succinct representation As far as we are aware, explicit single orbit generators of low weight have only been known for families such as the Hadamard code and Reed-Muller codes. For these codes the explicit description is to some extent obvious: the tests are supported on affine subspaces, or on evenly spaced points on a line ([33, 89, 7, 63, 69]). We only show explicit single orbit generators of low weight in BCH codes of design distance 5 (see Chapter 5). The novelty of this result lies in the fact that the support of a single orbit generator under AGL(1, 2n) can be described by a fixed set of carefully chosen univariate polynomials. This is a somewhat surprising uniform description of such codes, as n grows. While this result is a modest contribution, we believe that the question of finding fully explicit generators for BCH codes of general design distance could open an interesting direction of further investigation. Previous works Counting arguments using MacWilliams identities can show that BCH codes of small design distance must contain small weight codewords. While BCH codes are well-studied, only recently Kaufman and Litsyn [68] showed that these codes have a basis of almost smallest weight codewords. Such a basis was however arbitrary and unstructured and would require O( 2 n) bits to specify. Our result gives a basis that require only 0(n) bits to specify (i.e. the support of the generator of the single orbit characterization.) Sufficient conditions for structured testing In Chapter 6 we attempt to answer Question 3 above and provide some more general sufficient conditions for the existence of the single orbit property. We show that duals of families F C {F 2. -> IF2} that are invariant under affine transformations of IF2n and which contain a small number of codewords (namely, they are sparse,) must contain a small weight function that generates a basis (under the action of the affine group). See Figure for an illustration of these families in a broader hierarchy. Hence, we exhibit a general class of affine invariant families that admit structured testing.

23 We note that we can only show our result in some restricted settings of n, which are implied by the number theoretic machinery that we make use of. We also consider families that are cyclic invariant. Here we also show that duals of sparse families that are invariant under the cyclic group must have a low weight single orbit generator. Since a set of cyclic permutations is about a factor of 2" smaller than. a set of affine permutation of a given word, this is a somewhat stronger result. However, the settings of n for which it holds are more restrictive than before. Settings of n Our results for affine invariant families hold when n is prime, a condition that ensures that FT, does not have non-trivial subfields, as required by Bourgain's number theoretic results that we employ. For the cyclic case, we additionally need that 2' - 1 does not have large divisors. This condition is satisfied in particular when 2' - 1 is a Mersenne prime. We believe that our results should hold regardless of these restrictions, however our approach could lead to more general settings only if the number theoretic tools we use can be generalized. 1.5 Organization and credits Credits This work has been written in collaboration with Tali Kaufman and Madhu Sudan. Chapter 4 appeared in [60]; Chapter 6 appeared in [61]; Chapter 5 has not appeared in published form. Organization In Chapter 2 we introduce some basic definitions that we will use throughout the thesis. In Chapter 3 we give polynomial descriptions of families that are affine and cyclic invariant. We continue with presenting a counterexample to the AKKLR conjecture in Chapter 4. We start our study of the single orbit property by considering explicit tests for the common codes Hadamard, Reed Muller and some BCH in Chapter 5. In Chapter 6 we present our general conditions for succinct representation, and finally we describe some open problems in Chapter 7.

24 ... (a) primals (b) duals Figure 1-1: Relations among the families considered in this work. In (a): the red/black sets represent non-testable/testable families, respectively. In (b): diagram of the duals of families in (a). The red families also correspond to vector spaces that do not have a basis of low weight vectors. T represents the counterexample to the AKKLR conjecture from Chapter 4. C is a sparse, affine invariant family discussed in Chapter Bibliographic notes Testing various properties The test for linearity [33] generated numerous works on improved analysis [20, 75, 62, 92, 95], and on further generalizations to higher degree polynomials. The initial results [89, 15, 14] dealt with field sizes larger than the degree. More recently, low degree polynomials have also been considered over small characteristic by Alon et al. [7], and for other field sizes smaller than the degree, in works of Kaufman and Ron [69] and Jutla et al. [63]. Many other results demonstrated improved parameters for low degree tests in various settings [13, 48, 50, 90]. In graph properties, [53] also opened up the way for a long list of notable works [2, 4, 9, 8, 34, 66, 55, 5, 35] that considered different models and a wide range of related questions. Another domain of interest has been in testing boolean functions. Testing dictatorships, juntas, monotonicity, sparsity are just a few examples in the vast literature in

25 the area [86, 32, 43, 82, 58, 52, 47, 1]. Property testing has also been intensely investigated in the setting of testing distributions. Some examples of properties considered there are uniformity, statistical distance of pairs of distributions or entropy [18, 17, 19, 97, 3]. see [51, 87, 45, 78, 88]. For detailed surveys on developments on these aspects, Symmetries and testing Assorted results Symmetry has been invoked rather implicitly in testing results before the work of Kaufman and Sudan [72]. Graph testing in the dense model is one area where necessary and sufficient conditions are well understood by now [5, 35]. An implicit feature of classes of graphs is invariance under vertex relabeling. Their group of symmetries could be in fact much larger. For example, the group of symmetries of the class of bipatite graphs includes invariances under vertex removal or edge removal. In general, a graph property is testable if and only if it is 'regularly reducible' - a notion that abstracts invariance under a large group of actions on these graphs. This observation has motivated the search for similar groups of symmetries in algebraic settings that could enable testing applications. Properties that are symmetric under data relabellings have been also considered in testing distributions. In [97] Valiant considers such questions as testing the entropy of distribution or testing closeness between distributions. Symmetry was studied in cyclic codes by Babai et al. [16] to show that no good cyclic codes are testable. Also Goldreich et al. [56] studied symmetric properties and showed bounds on the randomness complexity of testing these families. Kaufman and Sudan's results Symmetry has been singled out explicitly as a common characteristic of known algebraic testable families only recently by Kaufman and Sudan in [72]. There they focus on general algebraic families of function f : K m -+ F, where K, F are finite fields and F is a subfield of K. Any such function is

26 in fact a special type of m-variate polynomial over K, which takes values only in F. Affine/linear invariance restricts the monomials that may occur in these families, and understanding monomial structure eventually leads to pinning down characteristics of the dual generators of the family. The general question of focus is how does the existence of a low weight function relate to the existence of a low weight basis (characterization,) and furthermore to testing in linear or in affine invariant families? They show a few main results in this direction. 1. First, they prove that families F C {K m -+ F} whose duals admit a weight k single orbit under affine/linear transformations of the domain K m can be tested by tests of locality roughly k. 2. They then relate the existence of a low weight function to the existence of a low weight single orbit. In families that are invariant under affine transformations of the domain K m, the existence of a function of weight k implies the existence of a single orbit of locality at most g(k, 1KI) = (k. K12)Jl 2. Notice that this quantity is independent of m, and it is constant when both k and K are constant. This dependency was slightly improved by Lin et al. [79] to a quantity that is still exponential in Kl. 3. They also relate the existence of an arbitrary low weight basis (characterization) to the existence of a low weight single orbit. In families that are invariant under linear transformations of the domain Km a k-weight basis implies a g(k, 1KI)- single orbit (under linear transformations). This perspective unravels a unified view of case specific analyses for testing Hadamard and Reed Muller codes [33, 89, 7, 63, 69], since these families do admit single orbit generators of small weight. It also motivates our search for other families whose testability is owed to the property that a low-weight single orbit generates the dual.

27 Chapter 2 Preliminaries We start with some standard notation. [N] denotes the set {1, 2,..., N}. A finite field of q = pt elements is denoted by Fq, where p is the prime characteristic of the field; F* = Fq - {0}. A primitive element of a field Fq, denoted w, is such that Fq {o, 1, w, w 2,..., W- 2 }. The inner product of two vectors x, y E Fn is denoted n-1 by (X, y) = E xiyi. The inner product between two functions f, g : D -4 F 2 is i=1 (f, g) = Z f(x)g(x). For a non-negative integer s, let bwt(s) be the binary weight xed of s (i.e., if s = E si 2 ' then bwt(s) = Z si.) 2.1 Algebraic properties Algebraic properties An algebraic property is a collection of functions {f : D - where D and R are finite of infinite domains and ranges. Most often D and R are vector spaces over finite fields. In particular, D = K and R = F12 where K is some finite extension of F. As it is the case in Property Testing, we study families of algebraic properties indexed by n - oc, namely families Fn C {f : K"(n) 9 F12(n)}. We will often drop the subscript n whenever the family is clear from the context. In this thesis we focus on binary functions (R = F 2 ) and on domains of size 2n or 2n - 1. This representation will also be useful at describing error-correcting codes, our focus throughout this work. R},

28 Functions vs vectors A function f : D -- F 2 can be represented as a vector in the following sense. Consider a fixed order of the elements in D, say ei, e 2,..., e D and represent f by its evaluation table at these points, i.e. by the vector (f(ei), f(e 2 ),., f.. (eldi)). Similarly, given a binary vector of length N we view it as the evaluation table of a function f : D - F 2, with D = N. Throughout this work we will alternate between these two equivalent descriptions of codes and families of functions without further comment. Linearity A property F C {f : D -± F 2 } is linear if it satisfies the condition that, if f, g E F then f + g E F. A linear property is essentially a vector space over F 2. A basis for F is a set of functions (vectors) that generate it. Dual of a linear property One can associate with every linear property F another property, denoted the dual of F defined as follows: F' = {g : D -± F 2 1 (f, g) = f ()g(x) = 0 for all f E F}. xed Duals of linear properties are extremely useful objects in the context of property testing. One is interested in particular in understanding their structure in terms of functions of small weight. Weight The weight of a function f : D - F 2 denoted Wt(f) {x E DIf(x) / 0}. The relative weight is defined as wt(f) = wtf Hamming distance The Hamming distance between f, g E F is defined as Wt(f -g). Similarly, the relative Hamming distance between f, g is wt(f, g). For a function g : D -+ F 2 and property F = {fff : D -a F 2 } the distance from f to F is Dist(f, F) = minge.f Wt(f - g), and the relative distance from f to F is dist(f, F) minger wt(f - g). We will most often use the notion of relative distance.

29 2.2 Error-correcting codes Typically, an error-correcting code C over an alphabet E is defined as the image of a function C that encodes a message m E E' into a codeword C(m) E EN. The alphabet E is usually a finite field F, or a vector space over F. In particular, one can now define the distance of C by A(C) = minci,c 2 Ec Wt(ci - c 2 ) and its relative distance by 6(C) = minc 1, 2 EC wt(ci - C 2 ). The dual of a linear code C was also implicitly introduced in the previous section as C' = {y (y, c) = 0 Vc E C}. 2.3 Property testing and locally testable codes We begin by defining 2-sided local testers, and then a 1-sided strong testers. The former testers are used in our negative results, while the latter in the positive sufficient testing conditions. Definition 4 (k-local tester) For integer k and reals E 2 > E1 > 0 and 6 > 0, a (k, E1, E2, 6)-local test for a property F is a probabilistic algorithm that, given oracle access to a function f E F, queries f on k locations (pro babilistically, possibly adaptively), and accepts f G F with probability at least 1 - c, while accepting functions f that are 6-far from F with probability at most Property F is called (k, 61, 62,6)- locally testable if it has a (k, ei, 62, 6)-local test. Given an ensemble of families F' ={ F}, we say F' is k-locally testable if there exzist 0 < 61 < E2 and 6 > 0 such that for every n, F is (k, 61+0(1), 62-0(1), 6)-locally testable (where the o(1) term goes to zero as n -- oo). We will often drop the subscript n when this is clear from the context. If both 61 > 0 and 62 > 0 then the tester is double-sided, else it is single sided. A single-sided tester is perfect if it accepts every function f E F with probability 1.

30 A tester is called adaptive if the queries it makes are based on answers to previous queries. Otherwise, namely when it can send all its queries at once, it is called non-adaptive. Definition 5 (strong k-local tester [57]) For integer k and real a > 0, a (k, a)- strong local tester for a property F is a probabilistic algorithm that, given oracle access to a function f G F, queries f on k locations (probabilistically, possibly adaptively), and accepts f E F with probability at least 1, while accepting functions f that are 6-far from F with probability at most 1 - a - 6(f, F). F is said to be strongly locally testable if there exist k < oc and a > 0 such that F is (k, a)-locally testable. Accordingly, we can define weak and strong Locally Testable Codes. Definition 6 (Locally Testable Code-weak version) An error-correcting code C is (k, ei, 62, 6)-locally testable for some integer k and reals E 2 > E1 > 0 and 6 > 0, if there exists a (k, El, 62, 6)-local tester for the property C. Definition 7 (Locally Testable Code- strong version) An error-correcting code C is (k, a)-locally testable for some integer k and real a > 0, if there exists a (k, a)- local tester for the property C. 2.4 Invariance Let F C {f : D -+ IF 2 } be a family of binary functions and let G C {r : D -4 D} be a set of transformations of the domain. A transformation w : D -± D is a permutation if it is a bijection. We say that F is invariant under G if for every f E F and every w E G it is the case that f o 7 E F, where for every x E D f o w(x) - f(7(x)). Automorphism group The automorphism group of family/code F, denoted Aut(F), is the group of all transformations 7 : [D] - [D] such that if c E C then c o - E C.

31 We are interested in families that are invariant under some well-studied groups (i.e., whose invariant groups contain some well-studied groups). In particular we look at invariance under linear and affine groups, defined over a domain of size p'. If eln then Fp'n Fne under addition, and one can consider a series of nested linear/affine groups by viewing Fp, as a vector space over Fe for all such e. Most generally, the linear group acting on a domain of size pf when the domain is seen as a vector space of dimension n/e over the subfield Fpe is n/e-1 GL(n/e, pe) ={7 : Fpn + Ipn 7F(x) = aizx", Vai E IFn}, i=o (see for e.g. [28].) A linear function w : F ' - F- n x) =Z() aix"" is a permutation if and only if the ai's involved above are linearly independent over Fpe (For a formal proof see Chapters 4 and 7 of [80].) Similarly, n/e-1 AGL(n/e, p') = {w: Fp, + IFpn1(X) = 3 aixz' + b, Vai, b E Fpn}, is the affine group acting on a domain of size p" when the domain is seen as a vector space of dimension n/e over the subfield Fe. These definitions are equivalent to the typical definitions of GL and AGL involving matrix transformations. We choose these descriptions since they provide a concise representation of all the linear/affine groups over a fixed domain size p". Observe that if eie 2 n then i=o GL(1,p") C GL(n/(eie 2 ), p12) C GL(n/ei, p") C GL(n, e)

32 and similarly AGL(1,p") C AGL(n/(eie 2 ),2 1 2 ) C AGL(n/ei, 21) C AGL(n, p). In this work we focus on algebraic families that are invariant under the smallest such linear/affine subgroups for p = 2, namely on GL(1, 2") and AGL(1, 2n), respectively. In fact, the linear families that are invariant under the largest linear/affine group over a domain of size 2n (i.e. GL(n, 2) and AGL(n, 2)) are well studied families of codes, namely variants of Reed Muller codes [42, 72]. Definition 8 (Affine invariance) A function r : lf2n -± F 2 n is an affine permutation if there exist a G F* and O3E Fn such that wr(x) = ax + b. A code C C JFN is said to be affine invariant if the automorphism group of C contains the affine group AGL(1, 2) ={7r(x) =ax + b, a E Fn, c F 2 n}. Linear invariance under the group GL(1, 2 n) is sometimes called cyclic invariance, due to the fact that the codes invariant under this group have a cyclic structure. We stick with this nomenclature hereafter. Definition 9 (Cyclic/linear invariance) A function 7r : F*,, -4 F* is a cyclic permutation if it is of the form 7(x) = ax for a E Fin. 1 A code C C JF-1 is said to be cyclic invariant (or simply cyclic) if the automorphism group of C contains the linear (cyclic) group GL(1, 2n) ={(X) = ax, a F*.}. 1 Note that this is a permutation of F*. if the elements of F3. are enumerated as (w, w 2 where w is a primitive element of F*22. N-1

33 Chapter 3 Description of Cyclic/Affine Invariant Linear Families Cyclic/affine-invariant families admit nice representations as sets of univariate polynomials. This description will be useful in our results and we start by making this connection explicit. In the last part of this chapter we relate this description to the classical representation of cyclic codes, as ideals in univariate rings of polynomials. Notation Let N = 2' and we view elements c E IF as functions c: FN -4 12, and similarly we view elements c E FN- as functions c : F*, - F 2. For d E {1,..., N-2}, let orb(d) ={d, 2d (mod N - 1), 4d (mod N - 1),..., 2"-d (mod N - 1)}. Also for a set D C [N] let orb(d) = UdEDorb(d). Notice that 2'd = 2id (mod N - 1) iff 2i(2i-- - 1)d = 0 (mod 2' - 1), and recall that 2-1 2" - 1 iff f n. Therefore, whenever n is prime lorb(d) = n for all d, and otherwise there exists d such that lorb(d) < n. Let lorb(d) = d. Let min-orb(d) denote the smallest integer in orb(d), and let D = {min-orb(d) Id {1,..., N - 2}} U {N - 1}.